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Keywords = Clarke’s tangent cone

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17 pages, 619 KiB  
Article
Optimality Conditions for Group Sparse Constrained Optimization Problems
by Wenying Wu and Dingtao Peng
Mathematics 2021, 9(1), 84; https://doi.org/10.3390/math9010084 - 1 Jan 2021
Cited by 1 | Viewed by 3578
Abstract
In this paper, optimality conditions for the group sparse constrained optimization (GSCO) problems are studied. Firstly, the equivalent characterizations of Bouligand tangent cone, Clarke tangent cone and their corresponding normal cones of the group sparse set are derived. Secondly, by using tangent cones [...] Read more.
In this paper, optimality conditions for the group sparse constrained optimization (GSCO) problems are studied. Firstly, the equivalent characterizations of Bouligand tangent cone, Clarke tangent cone and their corresponding normal cones of the group sparse set are derived. Secondly, by using tangent cones and normal cones, four types of stationary points for GSCO problems are given: TB-stationary point, NB-stationary point, TC-stationary point and NC-stationary point, which are used to characterize first-order optimality conditions for GSCO problems. Furthermore, both the relationship among the four types of stationary points and the relationship between stationary points and local minimizers are discussed. Finally, second-order necessary and sufficient optimality conditions for GSCO problems are provided. Full article
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18 pages, 332 KiB  
Article
Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets
by Stanisław Migórski and Long Fengzhen
Mathematics 2020, 8(10), 1824; https://doi.org/10.3390/math8101824 - 17 Oct 2020
Cited by 3 | Viewed by 2095
Abstract
In this paper, we study a class of constrained variational-hemivariational inequality problems with nonconvex sets which are star-shaped with respect to a certain ball in a reflexive Banach space. The inequality is a fully nonconvex counterpart of the variational-hemivariational inequality of elliptic type [...] Read more.
In this paper, we study a class of constrained variational-hemivariational inequality problems with nonconvex sets which are star-shaped with respect to a certain ball in a reflexive Banach space. The inequality is a fully nonconvex counterpart of the variational-hemivariational inequality of elliptic type since it contains both, a convex potential and a locally Lipschitz one. Two new results on the existence of a solution are proved by a penalty method applied to a variational-hemivariational inequality penalized by the generalized directional derivative of the distance function of the constraint set. In the first existence theorem, the strong monotonicity of the governing operator and a relaxed monotonicity condition of the Clarke subgradient are assumed. In the second existence result, these two hypotheses are relaxed and a suitable hypothesis on the upper semicontinuity of the operator is adopted. In both results, the penalized problems are solved by using the Knaster, Kuratowski, and Mazurkiewicz (KKM) lemma. For a suffciently small penalty parameter, the solution to the penalized problem solves also the original one. Finally, we work out an example on the interior and boundary semipermeability problem that ilustrate the applicability of our results. Full article
(This article belongs to the Section C1: Difference and Differential Equations)
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